The answer is no.
While the methods you learn in calculus like partial fractions, integration by parts, etc. are going to be limited, there is a complete algorithm, called the Risch algorithm for integrating elementary functions. Complete means that for any elementary function, the algorithm will either find an elementary antiderivative, or prove that none exists (note that an antiderivative always exists, but it will not always be an elementary function, for instance, it's well known that $\int e^{-x^2}\, dx$ is not an elementary function).
Since the derivative of an elementary function is always an elementary function, the Risch algorithm applied to any derivative of an elementary function will always work. It may not produce exactly the same function back, since the antiderivative is only defined up to an additive constant (i.e., you might get your original function back plus $C$).
Note that the Risch algorithm itself is very complicated, and involves some deep algebra, and a ton of different technical cases. It's much more complicated than the tricks you learn in calculus.
Many computer algebra systems implement some subset of the Risch algorithm. For instance, Mathematica (or Wolfram Alpha) should be able to integrate just about any derivative of an elementary function that you throw at it.
If you're interested in a high level view of the Risch algorithm, I recommend Manuel Bronstein's Symbolic integration tutorial. If you want a more detailed view (with pseudocode to implement in a compute algebra system), I recommend his book, Symbolic Integration I: Transcendental Functions.
You don't need to hit zero to stop, instead, you just stop when you know how to integrate the product of the two cells in the last row, i.e., it is simple enough. Then you can use the Tabular Method as the last step of it is to plus/minus the integration of the product.
For example, in your case, $\int(1/x * x)dx$ is time to stop, and $\int (e^x *(-\cos x))dx$ is good too because it is the negative of what we want to calculate and then we just need to solve a simple equation.
See more about the method:
https://en.wikipedia.org/wiki/Integration_by_parts#Tabular_integration_by_parts
Best Answer
Clearly $\int_u^v A'(t)B(t)dt=A(v)B(v)-A(u)B(u)-\int_u^vA(t)B'(t)dt$.